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Fearless Symmetry by Robert Gross, Avner Ash

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Here we explain how to describe some very important
elements of the absolute Galois group G. For each prime
number p there is a set of elements of G called Fro-
benius elements at p. As usual in mathematics, they are
important partly because we can describe them with some
explicitness, and partly because we can do something
interesting with them. What we will do with them in this
book is to make generalized reciprocity laws, beginning in
the next chapter.
Frobenius elements are difficult to define precisely. We
give an explanation in this chapter of a working definition.
The completely correct characterization is presented in a
(probably opaque) appendix to this chapter, so you can
see what it is like. We also discuss “ramification” in
this chapter; another appendix gives added details. These
appendices briefly define many concepts whose detailed
explication would form a large chunk of a text on algebraic
number theory.
Something for Nothing
Ferdinand Georg Frobenius was the nineteenth-century German
mathematician who invented the method of using characters to
178 CHAPTER 16
study group representations.
1
Today, his name is used for (among
other things) particular elements in the absolute Galois group G,
and also their restriction to the Galois groups of various polyno-
mials, that is, for particular permutations of the roots of various
Z-polynomials.
Whenever you can get something for nothing in mathematics,
you take it. Of course, “nothing is a relative term. What we
mean in this case is that we can define certain elements of every
Galois group by using an easily applied general theorem that tells
us significant information about them. These are the Frobenius
elements.
To take a simpler example first, let L be the field Q(f )forsome
Z-polynomial f . We know that L is composed of certain complex
numbers. Let τ be complex conjugation.
2
Then τ defines an element
of the Galois group G(f )ofL:Ifx + iy is any number in L, τ (x + iy)
is again a number in L,andτ respects addition, subtraction,
multiplication, and division. Also, we have a formula for it: τ (x +
iy) = x iy. Thus, τ is always there, and it is an important element
of the Galois group G(f ). It is the neutral element if and only if
L actually contains only real numbers, because then L does not
contain any numbers of the form x + iy with y = 0. W e can make τ
into an honorary Frobenius element at infinity, and call it Frob
.
This is purely a notational convention, and does not help us to
define Frob
p
when p is a prime. However, this notation Frob
is
common in the research literature.
The other Frobenius elements are at p for each prime p,and
we now begin to define them. If you choose to skip the rest of this
discussion, you need to know only:
Frob
p
is a particular element of G.
Actually, this is a lie, because we cannot define Frob
p
so
precisely. Really, given p, there is defined a certain
conjugacy class in G (depending on p), and Frob
p
is taken
1
In fact, Frobenius deserves credit for inventing group representations themselves.
2
We have used the letter τ before to stand for various things, but here we will use
it for the specific function of complex conjugation. We have been using c for complex
conjugation, but from now on we prefer Greek letters for elements of Galois groups.

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